To determine which binomial is a factor of the polynomial 9x² + 643x + 8, we first need to factor the expression. Factoring polynomials can often involve finding two numbers that multiply to a specific value, while also adding up to another value.
In this case, we’re looking for factors of the form (ax + b)(cx + d) that yield our original polynomial when multiplied together. First, we will calculate the product of ac (the coefficient of x²) and d (the constant term), which is 9 * 8 = 72. We are seeking two numbers that multiply to 72 and add to 643.
However, for large coefficients like 643, simple inspection may not immediately reveal factors. Hence, we can utilize the Rational Root Theorem or synthetic division to test potential rational roots. Using these methods, we can try values of x that might simplify our equation.
After testing various values or performing polynomial long division, we find that the binomial factor we arrive at is (9x + 1). Therefore, (9x + 1) is a factor of 9x² + 643x + 8.
For more complex polynomials like this, it might also be valuable to use a graphing tool or a software package that specializes in polynomial factorization to ensure accuracy and efficiency in finding the correct factors.